Sampling bias

The main source of bias is the Multiple Coulomb Scattering (MCS) in the material between TOF1 and TOF2. The particles used in the alignment go through 15 planes of scintillating fibre in each tracker, the aluminium windows of the spectrometer solenoids and focus coil absorber module and the helium contained in the bore of the magnets. In the absence of MCS, particles go down a straight path between the two detectors and are sampled in the two trackers in a fully non-stochastic fashion.

MCS introduces straggling and an uncertainty on the exact path taken by a particle between the two TOFs. Assume a perfectly Gaussian sample of particles first measured at a givenz0. In one dimension, the true position is distributed as x0 ∼ N(x, σ²_x) and the true gradient is distributed as x⁰₀ ∼ N(x⁰, σ²_x0). With material at z0, the true distributions at a more downstream positionz1 are

x1∼ N x+x⁰z01, σ_x²+σ_x²0z₀₁² +θ²_0,0z₀₁² , x⁰₁∼ N x⁰, σ²_x0 +θ²_0,0

, (5.12)

with z01=z1−z0. An expression for the RMS scattering angle atz0,θ0,0, can be found in [214]:

θ0,0=13.6 MeV βpc

q

dz₀/X0,0[1 + 0.038 ln(dz₀/X0,0)], (5.13) withβ =v/candprespectively the particle velocity and momentum and dz0/X0,0the thickness of the material encountered atz0in terms of radiation length,X0,0.

At each position zi where the sample encounters material, an RMS scattering angleθ0,i is accounted for. At the final measurement point, zn, this yields

xn∼ N

x+x⁰z0n, σ²_x+σ_x²0z²_0n+Pn−1

i=0 θ²_0,i(zi+1−zi)² , x⁰_n∼ N

x⁰, σ²_x0+Pn−1 i=0 θ_0,i²

, (5.14)

with z_0n = z_n −z₀. A non-zero mean gradient, x⁰, moves the position distribution byx⁰z_0n, and a spread in gradient translates in a growing spread in position. MCS smears both distributions but preserves the mean. The

RMS effective scattering angle,θ₀, and effective distance, ∆z, are defined as

The residuals between the measured and extrapolated distributions are distributed as

x_n−xⁿ₀ =x_n−(x₀+x⁰₀z_0n)∼ N 0, θ₀²∆z² ,

x⁰_n−x⁰₀∼ N(0, θ₀²), (5.16) withxⁿ₀ the track atz₀ propagated toz_n. The only source of uncertainty on these residuals is the MCS.

Assume that both distributions are measured atz₀andz_n.x₀is measured correctly butx_nhas a non-zero misalignmentδx_n. The mean residual between the two measurements reads

hxn−δxn−xn0i=hxn−(x0+x⁰₀(zn−z0))i −δxn =−δxn. (5.17) True distributions are inherently unbiased in the measurement ofδxn. The problem comes from the sampling of the distribution. A tracker station’s fiducial surface is circular and about 30 cm in diameter. A typical sample in the MICE beam has a spread of ∼80 mm in the upstream tracker and

∼200 mm in the downstream tracker. The sample is not contained within the trackers and hence the sample mean is not the true mean.

With limits of the sampling defined as [−xL, x_L], the sample mean of a Gaussian distribution centred aroundxand with a spreadσ_x reads:

ˆ

The relative deviation of the sample mean, ˆx, in equation5.18from the real distribution mean,x, is drawn in figure5.2as a function of the relative true mean and the relative true width. The sample mean preserves the true mean as long as the beam is well contained within the detector fiducial.

0 0.2 0.4 0.6 0.8 1 x)/x-x(

0 0.2 0.4 0.6 0.8 1 1.2 1.4

/xL

x 0.2

0.4 0.6 0.8 1 1.2 1.4

L/xxσ

Figure 5.2:Fractional discrepancy between the sample mean, ˆx, and the true mean, x, as a function of the mean and width of the true distribution in units of the range half-width,xL.

At different limits, the formula behaves as expected:

x→0lim xˆ= 0 =x,

xLlim

σx→∞ xˆ=xerf(+∞)−q

2

πσ_x lim

z→+∞

h

e^−z2sinh(z)i

=x,

xLlim

σx→0 xˆ= ^x₂ erfh

√−x 2σx

i−erfh

√−x 2σx

i = 0.

(5.19)

For a distribution centred at 0, the sample mean converges to the true mean.

A distribution contained within the detector, i.e. for whichσx/xL1, has a sample mean identical to the true mean as well. The sample mean of a broad distribution, i.e. for whichσx/xL1, is asymptotically zero and thus heavily biased, regardless of the true underlying mean.

Given a significantly off-centre mean of the distribution atz_n, the distri-butions in equation5.16are asymmetrically sampled. Given a positive mean, x_n, a particle that scatters towards the positive xis more likely to scatter out of the detector’s fiducial volume. The probability density function of the

effective scattering angle for a particle that is expected to hit xn atzn is

• ∆z, the effective distance between the two measuring stations;

• θ' ^xn0_∆z^−xn, the effective scattering angle;

• θ0, the RMS effective scattering angle as described in equation5.13;

• 1_[a,b](·), the indicator function that takes value 1 for all elements of [a, b] and 0 otherwise;

• C_n, the normalization constant so thatR+∞

−∞ f_xn(θ)dθ= 1.

The indicator function symbolizes that, for a scattering angle above threshold, the particle scatters out of the fiducial and does not contribute to the PDF.

The normalisation constant simplifies to Cn = 2 equation5.21is obtained using the approximation erf(z)'tanh(z).

The function in equation5.20is represented for different values ofxn/xL

in the left panel of figure5.3. At largexn, the function, fx_n(θ), is highly asymmetric due to scraping out of the fiducial volume. The choice of pa-rameters is consistent with a simulated 280 MeV/c beam travelling between the two trackers in the MICE Step IV configuration. The effective RMS scattering angle isθ₀'9 mrad and the effective distance is ∆z/x_L'17.5.

The average over the fiducial distribution ofx_n,g(x_n), reads f(θ) = C

−3 −2 −1 0 1 2 3

Figure 5.3:(Left) Part of the total scattering angle distribution (black) that does not scrape out for different extrapolated position at the n^th station,xn. (Right) Deformation of the total sampled scattering angle distribution (black) for different true position mean,xn, at the n^thstation.

withC⁻¹=R+∞

−∞ f(θ)dθ. This shows a deviation from the non-biased scatter-ing distributionN(0, θ₀²). In the limitxL/σx_n→+∞, the normal distribution is recovered.

The function in equation5.22is represented for different values ofxn/xL

in the right panel of figure5.3. For positive means,xn, the scattering angle distribution, f(θ), is biased towards negative values due to anisotropic scraping. Atxn = 0, the distribution is sharper than the normalN(0, θ₀²) and has tamed tails but retains the same mean. The large values of scattering angles are under-represented in the distribution due to inevitable transmission losses between measurement stations. The choice of parameters is motivated by the same requirements as for the other panel of figure5.3. The width of the distribution ofx_n is chosen to beσ_xn/x_L= 0.5.

At different limits, the mean of the formula behaves as expected:

xlim_n→0 hf(θ)i=R+∞

A centred beam withxn= 0 produces an even distribution inθ. A detector large enough to contain the entire beam also yields an unbiased scattering angle distribution. A pencil beam still produces an angular distribution sensitive to the extrapolated position at the n^th sampling station asθ0>0.

Figure5.4 represents the sample mean scattering angle, ˆθ, normalized by the true effective scattering angle,θ0, as a function of the true position mean,xn, and width,σx_n. The figure is represented for negative values of xn to produce log-friendly positive values of ˆθ/θ₀.

Provided that ˆθ6= 0 for most beam settings, equation5.17in terms of